Convergence to stable laws and a local limit theorem for stochastic recursions

• Autorzy: Mariusz Mirek
• Języki publikacji: EN

Abstrakty

EN

We consider the random recursion $Xₙ^{x} = MₙX_{n-1}^{x} + Qₙ + Nₙ(X_{n-1}^{x})$, where x ∈ ℝ and (Mₙ,Qₙ,Nₙ) are i.i.d., Qₙ has a heavy tail with exponent α > 0, the tail of Mₙ is lighter and $Nₙ(X_{n-1}^{x})$ is smaller at infinity, than $MₙX_{n-1}^{x}$. Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $Sₙ^{x} = ∑_{k=0}^{n} X_{k}^{x}$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.

Kategorie tematyczne

• 60F05: Central limit and other weak theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2010
• Tom: 118
• Numer: 2
• Strony: 705-720

Daty

wydano

2010

Twórcy

autor

Mariusz Mirek

• Institute of Mathematics, University of Wrocław, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland

Identyfikatory

DOI

10.4064/cm118-2-21